The American Dream is an interesting concept and ideal. The idea that if you work hard and are talented, you can rise as far as those qualities will take you in the United States might seem very similar to what people in most other countries want: personal security and individual success. We tend to think that in America, there is more opportunity to achieve this than in other places, making the American Dream unique.
In Steinbeck's Of Mice and Men, Lennie, George and Candy's simple American Dream is owning their own piece of land, with their own simple home. They could farm and not worry about getting "canned". Curly's wife wants to escape to a city where she can pursue an acting career, or even just a husband who is kind to her. Crooks just wants to be seen as a human being and a man, treated as an equal. These seem so achievable in ordinary times, but the Great Depression was no ordinary time. The society and economy of the 1930s crushes all of their dreams, along with millions of others.
The beauty of the story (and the American Dream) is that the Dream is so stubborn and persistent. We won't/can't let it die. Even in the worst of times, there is that hope, and Americans in particular often pursue that dream no matter how much tragedy they endure.